Now here we come to an important distinction that is ignored in discussions of Thin Realism: The Axiom of Choice didn’t get elected to the club because it is beneficial to the development of Set Theory! It got elected only because of its broader value for the development of mathematics outside of Set Theory, for the way it strengthens Set Theory as a foundation of mathematics. It is much more impressive for a statement of Set Theory to be valuable for the foundations of mathematics than it is for it to be valuable for the foundations of just Set Theory itself!
In other words when a Thin Realist talks about some statement being true as a result of its role for producing “good mathematics” she almost surely means just “good Set Theory” and nothing more than that. In the case of AC it was much more than that.
If by ‘thin realism’, you mean the view described by me, then this is incorrect. My Thin Realist embraces considerations based on benefits to set theory and to mathematics more generally — and would argue for Choice on the basis of its benefits in both areas.
This has a corresponding effect on discussions of set-theoretic truth. Corresponding to the above 3 roles of Set Theory we have three notions of truth:
- True in the sense of Pen’s Thin Realist, i.e. a statement is true because of its importance for producing “good Set Theory”.
- True in the sense assigned to AC, i.e., a statement is true based on Set Theory’s role as a foundation of mathematics, i.e. because it is important for the development of areas of mathematics outside of Set Theory.
- True in the intrinsic sense, i.e., derivable from the maximal iterative conception of set.
Again, my Thin Realist embraces the considerations in (1) and (2). As for (3), she thinks having an intuitive picture of what we’re talking about is extremely valuable, as a guide to thinking, as a source of new avenues for exploration, etc. Her reservation about considerations of type (3) is just this: if there were conflict between type (3) and types (1) and (2), she would change her concept to retain the good mathematics, in set theory and in mathematics more broadly. (This happened in the case of Choice.)
A more subtle point, quite important to us philosophers, is that Thin Realism doesn’t include a different sort of truth. Truth is truth. Where the Thin Realist differs is in what she thinks set theory is about (the ‘metaphysics or ‘ontology’). Because of this, she differs on what she takes to be evidence for truth. So what I really meant in the previous paragraph is this: benefits to set theory and to math are evidence for truth; intrinsic considerations, important as they are, only aid and suggest routes to our accumulation of such evidence.
- Pen’s model Thin Realist John Steel will go for Hugh’s Ultimate L axiom, assuming certain hard math gets taken care of.
I don’t know what you intend to be covered by ‘certain hard math’, but I take it a lot has to happen before a Thin Realist think we have sufficient evidence to include V=Ultimate L as a new axiom.
As I understand it (I am happy to be corrected), Pen is no fan of Type 3 truth
I hope I’ve now explained my stand on this: none of these are types of truth; types 1 and 2 are evidence for truth; 3 is of great heuristic value.
I am most pessimistic about Type 1 truth (Thin Realism). To get any useful conclusions here one would not only have to talk about “good Set Theory” but about “the Best Set Theory”, or at least show that all forms of “good Set Theory” reach the same conclusion about something like CH. Can we really expect to ever do that? To be specific: We’ve got an axiom proposed by Hugh which, if things work out nicely, implies CH. But then at the same time we have all of the “very good Set Theory” that comes out of forcing axioms, which have enormous combinatorial power, many applications and imply not CH. So it seems that if Type 1 truth will ever have a chance of resolving CH one would have to either shoot down Ultimate-L, shoot down forcing axioms or argue that one of these is not “good Set Theory”. Pen, how do you propose to do that? Forcing axioms are here to stay as “good Set Theory”, they can’t be “shot down”. And even if Ultimate-L dies, there will very likely be something to replace it. Why should we expect this replacement for Ultimate-L to come to the same conclusion about CH that forcing axioms reach (i.e. that CH is false)?
I think it’s simply too soon to try to make any of these judgments.